In Theorem REF  regularity properties shared by all solutions of (REF ) and (REF ) are found. Notice that we cannot directly apply the arguments of [1]}, [2]} based on the external sphere condition considered in [3]} because of the absence of graph and star-shapedness assumptions on the admissible free-crystal regions.
As a byproduct of Theorem REF  and Proposition REF  given a configuration \((A,u)\)  minimizing (REF ) resp. (REF ), we can construct a configuration \((A^{\prime },u^{\prime })\in \mathcal {C}\)  which minimizes both minimum problems (REF ) and (REF ) such that \(A^{\prime }\)  is an open set with cracks coinciding in \(\Omega \)  with the jump set of the corresponding minimizing free-crystal displacement \(u^{\prime }\) , and boundary \(\partial A^{\prime }\)  satisfying uniform upper and lower density estimates. Furthermore, we also observe that, any connected component \(E\) 
of \(A^{\prime }\)  that do not intersect \(\Sigma \setminus \color {black}J_{u^{\prime }}\color {black}\)  (up to \(\mathcal {H}^1\) -negligible sets), must have a sufficiently large area, i.e.,
\(|E|\ge (c_1 \sqrt{4\pi } /\lambda _1)^2,\) 
